Inverse Source and Inverse Scattering Problems

遥感的基本特点是所有的信息都必须通过（电磁波或声波）介质进行传输。我们只观察到散射场，即穿过中间介质（可能是空白空间）后的波。最简单的数学模型是标量波动方程，它通过傅里叶变换分解为每个波数处的标量亥姆霍兹方程。倏逝现象限制了分辨率，因为它只允许有限维的波族以没有严重衰减的振幅到达传感器。该族的维度提供了有关波源大小的重要信息，而在不同的基中表示波所需的非零系数的数量（以不同点为中心的汉克尔函数展开）携带了有关波源位置的信息。我们正在开发利用这些信息的方法。由于源并非唯一由散射波决定，这一事实使情况变得复杂。我们通过寻找支持（携带）辐射该散射场的源的最小集合来解决这个问题，并且必须是辐射相同场的任何其他源的一部分。实例表明，一般情况下不存在这样的凸集，但存在满足两个条件的分离良好凸集的最小并集。这个项目的一个主要目标是找到一种算法来找到这种分离良好的凸集的并集，并给出可靠的估计，即为了保证单个源在数据中可见，不同大小的源之间必须相距多远。这种分析与许多其他逆散射问题有关。一个特别重要的应用是理解玻恩或单散射近似是完全散射问题的可靠替代品的情况。虽然所有逆散射技术的实践者都知道，两个散射体相距越远，它们之间的相互作用就越少，因此玻恩的近似效果越好，但目前的数学理论似乎得出了相反的结论。这种明显的矛盾是由于我们用来测量和比较波浪的标准。大多数数学上易于处理的规范都有一些特征，这些特征歪曲了某些物理原理。我们继续努力制定足够简单的规范，以便进行数学分析，但又忠实地反映物理现实。遥感实验在没有任何直接接触的情况下从远处收集物体的信息。一种水下声学传感器阵列，根据潜艇发动机发出的噪声寻找潜艇（声源）的位置，是被动遥感系统的一个例子。而声纳阵列则是主动遥感系统的一个例子，它可以根据回声的特性传输自己的波来定位潜艇。在这种情况下，阵列，而不是潜艇，是主要的来源，潜艇被称为散射器。逆源和散射问题是遥感的重要组成部分，寻找小点状源和散射体的算法对包括天线设计在内的技术发展产生了重大影响。该项目旨在通过以更准确地反映该算法在有限数据集上的工作方式的方式重新制定分析，弥合有效算法与理论之间的差距。

The essential feature of remote sensing is that all information must be transported through an intervening medium by (electromagnetic or acoustic) waves. We observe only the scattered field, the wave after it has passed through the intervening medium (which may be empty space). The simplest mathematical model is the scalar wave equation, which decomposes, via Fourier transform, into the scalar Helmholtz equation at each wavenumber. The phenomenon of evanescence limits resolution by allowing only a finite dimensional family of waves to reach the sensors with an amplitude that has not been severely attenuated. The dimension of that family gives important information about the size of the source of the wave, and the number of nonzero coefficients needed to represent the wave in different bases (Hankel function expansions centered at different points) carries information about the location of the source or sources. We are developing methods to use this information. This is complicated by the fact that the source is not uniquely determined by the scattered wave. We resolve this by seeking the smallest set that supports (carries) a source that radiates that scattered field, and must have been part of any other source that radiated the same field. Examples show that there is no such set in general, but there is a smallest union of well separated convex sets that satisfies both criteria. A major goal of this project is to find an algorithm to find this union of well separated convex sets, and give reliable estimates on how far apart sources of various sizes must be in order to guarantee that the individual sources will be visible in the data. This analysis is relevant to many other inverse scattering problems. A particularly important application is understanding the circumstances in which the Born, or single scattering, approximation is a reliable substitute for the full scattering problem. While all practitioners of inverse scattering techniques know that the farther apart two scatterers are, the less they interact, and therefore the better the Born approximation works, current mathematical theory seems to say the opposite. The apparent paradox is due to the norms we use to measure and compare the waves. Most mathematically tractable norms have features which misrepresent some of the physics. We continue to work to formulate norms which are simple enough to allow mathematical analysis and yet faithfully reflect physical reality.A remote sensing experiment gathers information about an object from a distance, without any direct contact. An underwater array of acoustic sensors that seeks to locate a submarine (the source) based on the noise radiated from the submarine's engine, is an example of passive remote sensing system. While a sonar array that transmits its own wave in order to locate the submarine based on the properties of the echo, is an example of active remote sensing system. In this case the array, and not the submarine, is the primary source, and the submarine is referred to as the scatterer.The inverse source and scattering problems are an essential part of remote sensing, and algorithms for finding small point-like sources and scatterers have had major impact on technological development, including antenna design. The project aims to bridge the gap between effective algorithms and theory by reformulating the analysis in a way that more accurately reflects the way this algorithms work with finite data sets.